(Solved): The mass of a nucleus of rutherfordium- 254 is \( 254.1001 \mathrm{u} \). Calculate the mass in \( ...

The mass of a nucleus of rutherfordium- 254 is \( 254.1001 \mathrm{u} \). Calculate the mass in \( \mathrm{GeV} \mathrm{c}^{-5} \) (c) In 1919, Rutherford produced the first artificial nuclear transmutation by bombarding nitrogen with \( \alpha \)-particles. The reaction is represented by the following equation. \[ \alpha+{ }_{7}^{14} \mathrm{~N} \rightarrow{ }_{8}^{17} \mathrm{O}+\mathrm{X} \] (i) Identify \( \mathbf{X} \). i) The following data are available for the reaction. Rest mass of \( \alpha \) \[ \begin{array}{l} =3.7428 \mathrm{GeV} \mathrm{c}^{-2} \\ =13.0942 \mathrm{GeV} \mathrm{c}^{-2} \\ =16.8383 \mathrm{GeV} \mathrm{c}^{-2} \end{array} \] \[ \begin{array}{ll} \text { Rest mass of }{ }_{7}^{14} \mathrm{~N} & =13.0942 \mathrm{GeV} \mathrm{c}^{-2} \\ \text { Rest mass of }{ }_{8}^{17} \mathrm{O}+\mathrm{X} & =16.8383 \mathrm{GeV} \mathrm{c}^{-2} \end{array} \] The initial kinetic energy of the \( \alpha \)-particle is \( 7.68 \mathrm{MeV} \). Determine the sum of the kinetic energies of the oxygen nucleus and \( \mathrm{X} \). (Assume that the nitrogen nucleus is stationary.)